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MATH1151 week10c3 - Week 10 Friday 11-12 Class Time allowed...

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Time allowed: 20 minutes. 1. (2 marks) Suppose P, Q, R are invertible matrices of the same size. (a) Simplify the following expression, giving brief reasons for each step : P 3 ( RP 2 ) 1 ( Q 3 R 1 ) 1 Q 2 Q 1 P (b) If det( R ) = 1 2 , det( P ) = 3 and det( P 2 Q 2 R 1 ) = 2 , then find det( Q ). 2. (2 marks) Three companies X, Y and Z re-insure with one another to share their risk: For each $1 that X pays out in claims, X receives $0.30 from Y and $0.10 from Z . For each $1 that Y pays out in claims, Y receives $0.20 from X and $0.10 from Z . For each $1 that Z pays out in claims, Z receives $0.20 from X and $0.10 from Y . In a particular year, the total cost to X (claims paid and amounts paid to Y and Z , less amounts received from Y and Z ) was $26 million. Similarly, the total cost to Y was $22 million and the total cost to Z was $20 million. Determine the total claims paid out by companies X , Y and Z . 3. (2 marks) Suppose that | ˜ a | = | ˜ b | . Show that ˜ a + ˜ b and ˜ a ˜ b are orthogonal. 4. (4 marks) Consider the matrix P = 1 p q p 1 q where 0 < p < 1 and 0 < q < 1. Define the matrix S by S = P I , where I
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